On Tue, 19 Mar 2013 22:36:42 -0700, William Elliot <firstname.lastname@example.org> wrote:
>On Mon, 18 Mar 2013, David C. Ullrich wrote: >> >> >Let (f,X) and (y,Y) be compactifications of S. >> >Assume h in C(Y,X) and f = hg. >> > >> >Thue h is a continuous surjection and when Y is Hausdorff >> >a closed quotient map. >> > >> >k = h|g(S):g(S) -> f(S) is a continuous bijection. >> >It it a homeomorphism? If so, what's a proof like? >> >> I doubt that it follows that k is a homeomorphism, >> although I'm not going to try to give a counterexample. > >I'll have to check this out, but isn't >k = fg^-1:g(S) -> f(S) a homeomorphism?
That seems clear. Don't know what I was thinking, sorry.
> > >> Assuming it doesn't follow, then k being a homeomorphism >> "should" be part of the _definition_ of the partial >> order on the class of compactifications that we're >> struggling to define here. >> >> >> >> >>